THE COPERNICUS OF GEOMETRY equal, but, without using this postulate, it is impossible to prove that AXY, BYX are right angles, although they look it. If we assume the parallel postulate we can prove that AXY, BYX are right angles and, conversely, if we assume that AXY, BYX are right angles, we can prove the parallel postulate. Thus the assumption that AXY, BYX are right angles is equivalent to the parallel postulate. This assumption is to-day called the hypothesis of the right angle (since both angles are right angles the singular instead of the plural 'angles' is used). It is known that the hypothesis of the right angle leads to a consistent, practically useful geometry, in fact to Euclid's geometry refurbished to meet modern standards of logical rigour. But the figure suggests two other possibilities: each of the equal angles AXY9 BYX is less than a right angle - the hypothesis of the acute angle; each of the equal angles AXY, BYX is greater than a right angle - the hypotJiesis of the obtuse angle. Since any angle can satisfy one, and only one, of the requirements that it be equal to, less than, or greater than a right angle, the three hypotheses - of the right angle, acute angle, and obtuse angle respectively - exhaust the possibilities. Common experience predisposes us in favour of the first hypothesis. To see that each of the others is not as unreasonable as might at first appear we shall consider something closer to actual human experience than the highly idealized 'plane' in which Euclid imagined his figures drawn. But first we observe that neither the hypothesis of the acute angle nor that of the obtuse angle will enable us to prove Euclid's parallel postulate, because, as has been stated above, Euclid's postulate is equi- valent to the hypothesis of the right angle (in the sense of inter- deducibility ; the hypothesis of the right angle is both necessary and sufficient for the deduction of the parallel postulate)* Hence if we succeed in constructing geometries on either of the two new hypotheses, we shall not find in them parallels in Euclid's sense. To make the other hypotheses less unreasonable than they may seem at first sight, suppose the Earth were a perfect sphere (without irregularities due to mountains, etc.). A plane drawn through the centre of this ideal Earth cuts the surface in a great